Zimmermann identities and renormalization group equation in dimensional renormalization

Nuclear Physics B 167 (1980) 261-284 © North-Holland Publishing Company

ZIMMERMANN IDENTITIES AND RENORMALIZATION GROUP EQUATION IN DIMENSIONAL RENORMALIZATION Guy BONNEAU Laboratoire de Physique Th~orique et Hautes Energies, Paris*, France

Received 16 November 1979

In the framework of dimensional renormalization, Zimmermann-like identities are shown: D being the space-time dimension and ~(x) a monomial in the fields and their derivatives, they give the decomposition of the "oversubtracted" normal product N[(4-D)~(x)] on "usual" normal products. These relations are expected to be essential in the study of anomalies in the dimensional scheme as will be shown in a further publication. Moreover, using similar techniques, a rigorous proof of the renormalization group equation in the minimal dimensional renormalization scheme is also given.

1. Introduction Z i m m e r m a n n i d e n t i t i e s h a v e b e e n e x t e n s i v e l y u s e d in t h e B P H Z m o m e n t u m s p a c e f r a m e w o r k [ 1 ] e s p e c i a l l y in o r d e r to d e r i v e r e n o r m a l i z a t i o n g r o u p [2] a n d field e q u a t i o n s [3] a n d to s t u d y a n o m a l i e s ( e n e r g y - m o m e n t u m t e n s o r [3], a s y m p t o t i c scale i n v a r i a n c e [4, 5], A d l e r - B a r d e e n a n o m a l y in Q E D [6]). T h e s e i d e n t i t i e s allow o n e to e x p r e s s an o v e r s u b t r a c t e d n o r m a l p r o d u c t of fields as a l i n e a r c o m b i n a t i o n of m i n i m a l l y s u b t r a c t e d n o r m a l p r o d u c t s of o t h e r m o n o m i a l s in the fields a n d t h e i r derivatives. In t h e last few years, d i m e n s i o n a l r e g u l a r i z a t i o n [7] has b e c o m e an essential tool in q u a n t u m field t h e o r y , e s p e c i a l l y for n o n - a b e l i a n g a u g e t h e o r i e s . In this f r a m e w o r k a d i m e n s i o n a l r e n o r m a l i z a t i o n p r o g r a m has b e e n d e f i n e d , e i t h e r in m o m e n t u m s p a c e [8, 9] o r m o r e c o m p l e t e l y b y B r e i t e n l o h n e r a n d M a i s o n in F e y n m a n p a r a m e t r i c s p a c e [10]. N o r m a l p r o d u c t s a r e d e f i n e d , field e q u a t i o n s a n d action p r i n c i p l e a r e shown. Of c o u r s e o v e r s u b t r a c t e d n o r m a l p r o d u c t s d o n o t s e e m p o s s i b l e to define! But we shall s h o w in this w o r k that, a t t a c h i n g a f a c t o r u = 4 - D ( w h e r e D is t h e s p a c e - t i m e d i m e n s i o n ) to a special v e r t e x of a g r a p h l e a d s to r e l a t i o n s v e r y similar to Z i m m e r m a n n identities. M o r e o v e r , the p r o b l e m of a n o m a l i e s in d i m e n s i o n a l r e n o r m a l i z a t i o n (trace a n d 3'5 a n o m a l i e s ) is closely r e l a t e d to such i d e n t i t i e s (see also p a g e 29 of ref. [10a]): this, as well as the p r e s e n t l a c k of Z i m m e r m a n n i d e n t i t i e s a n d * Laboratoire associ6 au CNRS. Postal address: Universit6 Pierre et Marie Curie, 4 place Jussieu, Tour 16, ler 6tage, 75230 Paris Cedex 05, France. 261

262

G. Bonneau / Zimrnermann identities

renormalization group in a dimensional framework, motivates our study. (A forthcoming paper will be devoted to the problem of anomalies [11].) Indeed, another yet unsolved problem in this field was the derivation of the renormalization group equation in the minimal scheme (without any normalization conditions) - if one excepts the usual "proofs" with infinite renormalization constants Z. This question was not considered in ref. [10] and in the work of Collins [8] it was simply asserted, without any proof, that the differential operation tz 20/O/z 2 on a Green function can be expressed as a linear combination of differential vertex operations (DVO's, see ref. [2]) in the same manner as in B P H Z momentum-space renormalization [2]. This assertion will be proved in this work using techniques very similar to the ones used for Zimmermann identities. Sect. 2 is devoted to the formulation of dimensional renormalization following ref. [10a]. Then, Zimmermann-like identities are proved in sect. 3 and the renormalization group equation in sect. 4. Appendices B and C consider the same problems in the momentum-space dimensional formulation of Collins and Speer [8, 9].

2. Dimensional renormalization: generalities and notations We shall use the formulation of dimensional renormalization given by Breitenlohner and Maison in ref. [10a] (hereafter referred to as BM). Very precise definitions of the algorithm (as well as the convergence proof) are given there, but we shall often use simplified notations [see comments (i) and (ii) in subsect. 2.3]. These and a summary of the main results will be given below.

2.1.

T h e forest f o r m u l a

Following the ideas of usual B P H Z method [1], a forest formula gives directly the renormalized amplitude R ° associated to a Feynman graph G: RG(p) - = lim ~-,o R ~ (P) - = lim ~ - ,m oilo(÷

R~(p)) ,

(1)

with o R~,~(p) =

--

, . , (p; g ) . da_ II (~ - C H ) I-G

Z (qg, o - ) o

((qLO-)G)

H ~ cg

Here: p = 4 - D , where D is the space-time dimension used to regulate the theory; e appears in the denominators of the propagators; (~, o')G is a labelled forest for the graph G: its definition and properties will be recalled in subsect. 2.2; a_ is the set of L F e y n m a n p a r a m e t e r s associated with each line Ii of G (~G = {l,, i = 1, 2 . . . . . L});

G. Bonneau / Zimmermann identities

263

~((~, or)G) is the integration domain for _a corresponding to the labelled forest

(~¢, Or)G; "G !~.~ (p, _a) is the dimensionally regularized amplitude associated with the graph G,_p being the set of its external momenta: the precise prescription for its evaluation is given in BM; Cn is the subtraction operator on 1PI subgraph H: its meaning will be schematically given in subsect. 2.3. In the product I-[H (1-CH) the order is defined with operators (1-CH) for smaller graphs acting first (on the right). Of course this ordering is not uniquely defined because the subgraphs H of the maximal forest ~ are not necessarily included into one another; they can also be disjoint but the result will be independent of the particular choice of ordering! This order allows their indexation Hi according to

i
Hi c H i HinHj=&.

(2)

2.2. Definition and properties of labelled forests Labelled forests (~, or)G have been introduced in BM: the maximal forests ~ will correspond directly to counterterms in the lagrangian.

Definitions ! (see appendix A of BM) Let G be a connected graph with 1PI components G1, GE. . . . . G,. A maximal forest for G is a maximal set of non-trivial, non-overlapping, 1PI subgraphs of G. For any maximal forest ~ and any H ~ ~ let/.t (H) be the set of all maximal elements of properly contained in H and let I:I be H//~(H). A labelled forest is a pair (~, or) consisting of a maximal forest c¢ and a mapping or: c¢-->~6 such that or(H)~ *L~'I:I= {lines of 17t}. ~ ( ~ , or) is the subset of the a-space given by ( ( ~ , Or ) =

{(O/1. . . . .

OraL): at > 0 Vl ; al --<--a~CH)for I e H s re}.

Properties I: (see lemma 3 of BM) (a) Any maximal forest ~ for G is a disjoint union ~ = @ i~=~~; of maximal forests ~i for the components Gi of G; Gi e ¢¢i. (b) Any maximal forest ~ for G can be labelled. (c) For any 1PI subgraph H of G, there is a natural one-to-one correspondence between labelled forests (c¢, Or) for G such that H ~ ~ and pairs {(~1, Or1), (~2, o'2)} of labelled forests for G / H and H. (d) Any maximal forest for G has exactly B (G) elements [B (G) = number of loops of graph G]. (e) G - o r ( R ) is a tree in G. (f) U(~,o-) ,~(~, or) = {al :a~t --> 0 Vl}. (g) For (c~, or) .~ (c¢,, or,), ~ ( ~ , or) A ~(c~,, or,) is a set of Lebesgue measure zero.

G. Bonneau / Zimmermann identities

264

2.3. The subtraction scheme We are now able to describe the subtraction scheme. In the following, because of the factorization of the amplitude for a connected graph into a product of amplitudes for its 1PI components (plus propagators for the lines connecting them!), we shall restrict ourselves to 1PI graphs (external lines are always removed leaving only external momenta). We now introduce for each labelled forest (c¢, o-)o the new variables (t,B)= (tH, H ~ ~ ;/3t, l ~ ~ = LPG\o'(~)) and auxiliary parameters ~H and ~H: = tH~H =

_q--> (_t, fl): a l =

H~

~t-I,

if l = o-(H), H e ~g,

~

(3)

(/31( 2 ,

if l e ~ h

= ~a\cr(H), He ~.

We then obtain R~ =

E

( R . ,o. ) ( ~ , ~ ) ~ ,

(~, o-)G

with (g~,,)(~,,~)c

c~

d/ZH, ( 9 - Ca,) I~G.,(p; t,/3),

(4)

and 1

(OG = co, OH= 1 for H # G) This equation needs a few comments. (i) The order between Hi's being defined by eq. (2), expression (4) means that one starts with the integrations over the parameters associated with the lines of HI:~ d/xrh. Because of the presence of a factor t~i~(rh)-°'H~ - - resulting from the change of variables (3) (where 0-)Ht is the superficial degree of divergence of graph Hi) - - which can be interpreted as a distribution 1 /~iB(H)_,or~ _ ( - - ) ~°H tH ~OH]

1 6('°H)(tH)+regular part at • vB(H)

0,

(5)

the result is a meromorphic function of v with poles at v = 0 (and elsewhere but we do not care). The singular part [first part of eq. (5)] can be proged to be a polynomial in the masses and external momenta of the subgraph. The operator - CHI then adds the counterterm for HI which is [see appendix B of BM and the appendix A of this work, especially formula (A.2)] proportional to the aforementioned singular part*, with a factor ~ iB(H') missing (this factor is the source of difficulties in the convergence proof * M o r e precisely, the c o u n t e r t e r m is the s i n g u l a r p a r t of the L a u r e n t s e r i e s a r o u n d u = 0 of this s i n g u l a r part.

G. Bonneau / Zimmermann identities

265

and comes from the difference in the number of loops of G and G/H~; in the same way there is a factor ~ ~B~H0 coming from the measure dt~H~). We continue in this way until the last overall subtraction (9 - CG). Let us introduce a special notation for the amplitude subtracted for all genuine subgraphs of G belonging to fig, tr)G:

[I

(6)

(9 - c . , ) } I2~(p; , _ t,

then G

--G

( R ~.~)~,,,)o = (9 - C G ) ( R ~,~)~,,~)o .

The more precise definition of the operator - Ca given in BM insures that one gets a correct B P H Z forest formula and validity of the field equations and action principle. (ii) In parametric space the integrand for G is not factorized in the integrands for Hi and G / H i respectively! In particular, external momenta of Hi are linear combinations of external momenta of G, associated with vertices of H , and internal momenta of G / H I that are not explicitly present in the integrand. So one needs an operator which permits the insertion of the counterterm for Hi, -CH,I~,~, H. into the amplitude for G / H I : this operator UH, has been constructed in ref. [10a], appendix B, and will be used to note the insertion of any polynomial in external momenta as a vertex function into the amplitude for a reduced graph. Comforted by this knowledge we shall also use in this work a schematic notation " © " to summarize the algorithm described in the first comment*: G~.~ (R)~,~)o

=

H,~\I~,

dtZH(~ -- '--H~j . . . .

'--'. . . .

r " "~]./'G/Hir",,tRH" ~

,

,

where the pair {(c~, o-~ )Z/H,, ( ~ , a'~)H,} is associated with the labelled forest (c~, o-)o containing the graph Hi (see property I(c)). (iii) In the m i n i m a l dimensional scheme one usually explicitly introduces a scale factor/z ~ for each loop integration [7, 12] (see also subsect. 4.1), that is, for each ti-i, integration**. The renormalization group equation that expresses the tt dependence of the G r e e n functions will be demonstrated in sect. 4. As this/z was not introduced in Breitenlohner and Maison's work, we have shown in appendix A that, with slight * We emphasize again that this is neither a product (nor an insertion) of the renormalized amplitudes for Hi by (into) the one for G/Hi. In particular some factors ~ia~ e~S(Hp should not be forgotten. ** Indeed an IZIcannot be a tree (the corresponding H would then be 1PR) and then has at least one loop; now, as there are B(G) of them (property I(d)), each I7I has exactly one loop.

G. Bonneau / Zirnmermann identities

266

modifications of the proofs, the fundamental results remain the same (proposition 3 and theorem 1 of BM) and we now give them. Theorem l ( a ) : For each labelled forest (oK ~r)c and associated pair ((c~1, o'1), (c~2, ~rz)) for G / H and H, respectively, the singular part of the Laurent series expansion of (R-Hv.~)t~2,=2) consists of poles of order B (H) or less and is a polynomial of degree wH in the masses and external m o m e n t a of the subgraph H. It vanishes if H is superficially convergent. Theorem 1 (b ): The amplitudes (Rv.~)l~,~l o - - and consequently R ~ - - are analytic at D = 4. The limit e ~ 0 + exists and is again analytic at D = 4; the value of R ~ for v = 0 is the renormalized amplitude for G. These theorems express the main useful properties of dimensional renormalization. One special feature of the minimal dimensional scheme (also called mass independent renormalization scheme [12]) is the independence of Ix of the singular part [no In (m2/ix 2) for instance]. Of course, if one wants to implement normalization conditions by adding finite counterterms (depending on /z and m) to the lagrangian as in refs. [8, 10], this last point no longer remains true; of course theorems l(a) and l(b) still stay with the understanding that the coefficients of the polynomial can depend on functions of m2/ix 2 that correspond to these finite counterterms. As a last remark, we recall that in ref. [10a] only massive particles are considered. The case of a theory with massless particles has also been resolved by Breitenlohner and Maison in ref. [10b] and we think it is clear that our derivations would also hold for this general case. 3. Z i m m e r m a n n identities The derivation will follow very closely the lines of the usual B P H Z one (refs. [1, 13]).

3.1. Lemma 1 We begin with the easily demonstrated lemma: Lemma 1 : t i p ) being a m e r o m o r p h i c function of v with poles at v = 0, [-p.p.

(.fO,))]-E-

×

-P.P. (f(v))] = r.s.p. (f(v)),

(7)

where p.p. (f(v)) means the singular part of the Laurent series f o r f ( v ) near v = 0 and r.s.p. (f(v)) the residue of the simple pole of f(v) at v = O.

3.2. Lemma 2 Let us consider a 1PI Feynman graph G (the extension to the general case of a connected graph being straightforward, see subsect. 2.3) and the associated reG normalized amplitude R ....

267

G. Bonneau / Zimmermann identities

We now consider the s a m e graph with a factor v attached to a special vertex V of G. Let ~78(x) be the corresponding m o n o m i a l in the fields and their derivatives and 8 its canonical dimension. W e want to compute the associated renormalized amplitude R ~ and show l e m m a 2. L e m m a 2: G ~

G

, - v R ~, , = Y, U , , (r.s.p. R ,,~

----3,t - ~ R

/~,. ~:~)/¢ G,.~ ',

(8)

3'i

where y~ are 1PI subgraphs of G containing V as one of its vertices (yi = G allowed). The notations Uv, a n d / ~ have been defined in sect. 2. (Uv, being an operator on im/v, is at the left of this reduced amplitude and we have shortened the notation v, e which would have been

"/i ( ~ , o ' ) G / v i

H~qg

W e shall do the same whenever there is no misunderstanding possible.) Let us consider a special labelled forest (~, o') for the graph G (or G~). We divide the family ~ into three forests: Fv: v~eFv¢:~ vie (~

and

V E V

i

FA: a ~e F A ¢* a ~~ ~, V g a ~ and FB: b ; ~ F B ¢~, b i ~ ~

and

,

a' c

Fv,

b iZ {Fv, FA} .

~,,)(~,~,), the first subtractions are associated with the forest FA When evaluating (R c~ and are unaware of the presence of the factor v; then, G~

G ~/aP

(R~.~)(~.,,)= (R~.~

ap

)(,~.~)~/ ~)(R~.~)(,, .... )o~,

where a p is the largest element of the forest FA. T h e n comes the subtraction - C ~ , ~ dtx~, where v ~ is the smallest element of Fv (elements of Fv are strictly ordered: v t c v 2 . . . c vq_=_G). H e r e the factor e is concerned, and with l e m m a 1 one gets (l~G./vl'~(~l

v~

-v~

G/o~

(Rv ~) + Uv~(r.s.p. (R ~,~)(~2,~2)o,)(R ~,~ )(~,~)o/~ •

Subtractions associated with elements b ~ of FB being unaware of the factor v, this can be iterated and finally we obtain ( R . G, ,~ ) ( , , . ) -

v ( R vG . , ) ( , . , ) = Z

•

V~v

-- v(

•

GIvi

Uv' (r.s.p. (R o'.,)(*2.¢2)v')(R,., )(*~.¢,~/~i •

vzEe~ i

(9) We now sum over all labelled forests for G. E v e r y 1PI subgraph yi (with V ~ %) being contained in at least one of the (~, o-)'s, the sum on the right-hand side can be

G. Bonneau

268

/ Zimmermann

identities

made on these y~ and we obtain

R~:-vR°~

Y~ U,,(r.s.p. "l'i : 1PI_~ G VC'yi

Y~

-*'

)

(qg2,o')~, i

o,,. (c~ 1,0"1 )O/,~i

(10) where the existence of a o n e - t o - o n e correspondence between { ( ~ l , O ' l ) G / 3 , p (rg2, o-2),,} and (rg, o')o containing ~/i insures that one obtains the complete set of labelled forests for G / y i and ?i on the right-hand side of eq. (10). This completes the proof of l e m m a 2.

3.3. Zirnmermann-like

identity

We now use a fundamental result of the renormalization program [theorem l(a)]: the singular part of the amplitude for y;, subtracted for all its 1PI genuine subgraphs, is a polynomial in the masses and external m o m e n t a of degree toy,. H e r e toy, depends on the canonical dimension 8 of the special vertex ~?8(x) [in (~4 theory, ~o,, = 4 - ni + i ( 8 - 4)]. Then we can express r.s.p. R-- 3,~., as /~1

r=0

{il,i ...... i,)

r!

ap~ ~

/'1"2

/~r

P i l Pi~ " " ' Pi, • api2

• • •

pl,

(11)

Pi s O

The pi are the external m o m e n t a for subgraph yi with ni external lines. We can rearrange the sum over subgraphs yi according to the n u m b e r n of external lines of subgraphs F~ of G containing the special vertex V: n =nma x n =nmin

t ° F ~ n 8)

Fi:VeFi~-G

r=0

n external lines

{il,...,i,) 1 --- ii _--
" Vi.')--... 'J.

\r.

{

r.s.p. Op~'

-

--F.

api","

R~;~

li

p,-o

(12)

nmin ---->2 in order for G to be a 1PI graph; nmax depends on the particular field theory and dimension 8 of the special vertex ~78(x) considered, and is the maximal n u m b e r of external lines for a graph with insertion ~78(x) to be superficially divergent. The insertion of ( 1 / r ! ) p ~ . . . . Pi"," will lead to the insertion of the normal product

r!

[k=l

Eq. (12) is graphically drawn in fig. 1.

269

G. Bonneau / Zimmermann identities

q

ffl.

Fig. 1. Graphical interpretation of a Zimmermann identity [eqs. (12), (13)].

We now sum over all graphs G contributing to the (01T(N[p•8(x)]X)[0) or°per and get, the limit v, e -* 0 being taken,

(01T(N[taffs(x)]X)10)or°per=

"m,,,,~8) ",~", ~) ~] E r/~2

r=O

E

{ ( -- i) ~ r!

l
r.s.p. (0[T(N[Ca(0)]~(px) × (0 T ( N [ 1 ~ 1

function

0~ .. 0 ~'~ OP~ 1"

{i . . . . . . it}

Green

Pit

• • • ~(p,))[0)°r°P°r]p,~0}

[(i~I~=k~t't')~)](X)] X)XO>prOper (13)

The combinatorial problem was exactly the same as the one of usual Zimmermann identities (see for example ref. [13] p. 124): this justifies the result (13). The tilde indicates Fourier transformed fields and the bar on the Green function means that the overall subtraction has not been done [see also eq. (6)].

G. Bonneau/ Zimmermann identities

270

Eq. (13) can, of course, be rewritten as

",n-(s),~o~,8) [(_i)' N[Ws(x)] =

~ n>2 r.s.p.

r=0

Z {i,,...,i,}

r!

0' Op~. . . .

l
0Pi,~'

(OlT(N[~78(O)]~(pl) " " C~(p,,))[O)P'°Perlp,.o]

xN[l

~ l [(i}-I=kO,a)¢](x)] ,

(14)

which is the desired Z i m m e r m a n n - l i k e identity*.

3.4. Examples in c194(x)/4! theory

~)4(X) (a)

~78(x)=--

~tov(n, 6 ) = 4 - n

6=4

4!

~ nmax=4.

Then,

(/)4(X)]

N[ v[~"j

= qN[½¢2(x)] +

rN[l(o,¢)2(x)]

1 2¢ ( x ) ] + sN[~cI)O

1 4

+ tN[~.,qb (x)],

(15a)

with q =r.s.p.

0 T N

-1 r = r.s.p. 4

s=r.s.p. r.s.p.( (b)

0

(0)

0

,

( 0 TI ( N [{~-~(0) ] ¢"( p ) ¢ ("q ) )[ 0 / prOper

0

Op, Oq~,

-- 10 4

(0)~(0)

40

Opt, Op~

] ([4.

~78(x) =~(0,,~)2(x):

p=q=o'

) 0~proper 10) l'r°per"

6 =4

~ too(n, 6) = 4 - n

~ nmax=4.

(15b)

O n e gets the s a m e f o r m u l a with N [ ~ 4 ( 0 ) / 4 ! ] changed to N[½(0~,~)2(0)] in the coefficients q, r, s, t. (c) ~78(x)=~lt~ 2(X): 6=2 ~tov(n, 6 ) = 2 - n ~nmax=2=nmin * In the minimal dimensional scheme, as said in comment (iii) of subsect. 2.3, the coefficients of normal products on the right-hand side are scale (/.Q and mass independent--except for the possible nai've mass dependence - - and then functions of coupling constants only. On the contrary, when normalization conditions are implemented by finite counterterms, that is to say new vertices, relation (14) still holds but the coefficients are no longer scale and mass independent.

G. Bonneau / Zimmermann identities

271

Then, N[~,½qb2(x)] = r.s.p. (0IT(N[½~2(0)]~(0)~(0))I0)pr°perN[½@2(x)].

(15c)

3.5. Generalization to Green functions involving normal products The calculation of (01T(N[u68(x)]I-[~'=l N[Qi(yi)]X)lO) pr°per will proceed along the same lines as above, the only difference being that there will exist different kinds of graphs 3'/that contain the special vertex V (depending on the number of normal products N[Qi (Yi)] inserted in this subgraph Yi). We shall not give the explicit result (see ref. [13] p. 125 for such a formula in the usual B P H Z momentum space subtraction framework) but rather indicate in fig. 2. the structure of the Zimmermann identity in the simplest case of one extra insertion N[Q( y)]. 4. Renormalization group in the minimal dimensional scheme

We shall say a few words about the introduction of the scale/z in subsect. 4.1, then prove the assertion of Collins that the differential operation i.L2(a/a~ 2)(01T(X)I0) can

!

Fig. 2. Graphical interpretation of a Zimmermann identity for a graph G with an insertion N[Q(y)].

(7. Bonneau / Zimmermann identities

272

be expressed as a linear combination of differential vertex operations (DVO, see ref. [2]) in the same manner as in the usual B P H Z case. In subsect. 4.3, we then obtain the renormalization group equation and lastly discuss its generalization to G r e e n functions with insertions (subsect. 4.4).

4.1. Introduction of the scale Iz It is impossible to introduce the scale/x in the lagrangian in a satisfactory way, which forbids the use of the action principle to get ~=(a/a~2)(01T(g)10). Indeed, if one takes as effective lagrangian* for ~ 4 theory (for instance) l_fa

t~X2

~fr(x)-2~ ~ j-~

lm2¢~2

~4 -g/~ 4!' v

(16)

the factors/.~ ~ that will appear in the amplitude should not be all developed near v = 0 in order that the G r e e n functions have the right dimensionality before the limit v ~ 0 is taken. Then

F2N(Pl . . . . .

P2N)regularized =/X ~(~r-l~/~2N(pl . . . . .

(17)

P2N)regularized ,

where the renormalization program is applied to /~2N. This can be said in other words: the power o f / z ~ should count the n u m b e r of loops B which differs from the ~4

number of vertices V: B = L i n t - V + 1 = V - (N - 1). Multiplying the whole lagrangian by a p o w e r / x - ~ would also be unsatisfactory because now this would count the number of loops minus o n e ( L i n t - V = B 1)! Then a prescription has to be done by hand and this was done in formula (4) where a tz ~ was introduced for each loop m o m e n t u m integration ~(d°k/(2zr)D)iz~ ~

f (dtH/tH)p. ~.

4.2. Differential vertex operation/~2(0/a/z2)(0[T(X)10) p r ° p e r 4.2.1. Let G be one of the F e y n m a n graphs of the G r e e n function (0]T(X)I0)Pr°per; the proof will be by induction with respect to its n u m b e r of loops B (G). Lemma 3: Let (~g, o-) be a labelled forest for graph G. Then, 2

Ix ~

19

G

1

G

(R~,~)(~.~) = ~vB(G)(R~,~)(~,~) + Z UH(B(H)×½r.s.p. (R-H~,,) (~2,~,)H)(R G/H ~,~ ) (~m,o'l)G/I-I "

H~

(18) * Let us recall that in the minimal dimensional renormalization (~tla Zimmermann) there is no need of any counterterm in the lagrangian.

G. Bonneau / Zimmermann identities

273

C o n c e r n i n g the notations, see also the c o m m e n t following l e m m a 2. With

foOH(l,t2) v/2 2dtH fox ti4 ,1-I~_ d/3~,

f d/.t,H =

-G

l e m m a 3 is easily p r o v e d for o n e - l o o p graphs: as R ~ = (~ - C o ) R ~,~ and - ( 1 / v ) p . p . R- ~~.~ i n d e p e n d e n t of/~ [ t h e o r e m l(a)], we get 2

0

G

OI.t

2

2 R ~ • =l~

0~

O~

-CGR.,~

--G

=

--G 1 --G I --G 1 --G 2R~e, = ~ v R . ~. = ~ v ( ~ - CG)R ,.~ + ~ v C ~ R ~.~

1 r)G

, 1

~G

= ~u~x ~,~ 1-i r.s.p . . . . .

Q.E.D.

L e t us n o w suppose that l e m m a 3 is true for graphs G with at m o s t N loops; consider a 1PI g r a p h F with N + 1 loops and any of its labelled forests (q¢, Cr)r. ~ (F) = {G1 . . . . . Gp} being the set of all m a x i m a l e l e m e n t s of c£ p r o p e r l y c o n t a i n e d into F, we have to c o m p u t e * 2 0 X =/A,

(~-Cr)

u~r .....

0)tt 2

~(R~,~)(~,~)o,

,

Gi

w h e r e (c~, o.)c~ are the labelled forests for the disjoint G i ' s (C~r = {F, {c¢i}}). In this minimal scheme x=/~

2

0

--02

{I~

,r/u(r)r-,~ o. } u/Zr~,,~ w llG,(R~,~)(~,~)~, .

(20)

W e n o w use the r e c u r r e n c e hypothesis ( B ( G ~ ) ~ N ) and get

x [21 vB (Gi)(R G~ ..~)(~.~,

+ ~

UH,,(B(Hi, ) .

HilE~i

1 x ~r.s.p.

1 --F =~vB(F)(R.,~)t~,,~)+ ~

t l ]{l~GiIHi r] I I (~l,tT2)Htt,k--v,e :(~l,~l)oi/Hi,] )

~,--/]~n'l~v,e,

U H ( B ( H ) × ~r.s.p.

-H

-F/H

HeC¢

H#r

(21)

M o r e o v e r (R F~.~)(~.~) has no poles at v = 0; then the s a m e is also true for x: 0 = - p . p . x = ½ u B ( F ) x - C r (R- r~,~)(~,~) + B (F) × ~r.s.p. 1 ~ (R- - I,,~)(~,~) + Z

-

C r { U H ( B ( H ) × ½ r . s . p . (Rv.~)oe -H -rm )(~,,~'l)rm}" .... )H)(Rv,,

HeR H#F

(22) * As said before the notation C) and I] is used for simplicity: in fact there is no product here except for the subtraction operators l]n(~- CH).

G. Bonneau / Zimmermann identities

274

Here lemma 1 has been used. We combine eqs. (21) and (22) and get the desired result of lemma 3: 1

F

x = ~ v B ( F ) ( R ~.~)(~.~)+ Y. UH(B(H) x ½r.s.p. (R~,~)(%.,,~).)(R~.~)(~ -u r/H .... )F/..

Hcqg

We now sum over all labelled forests for F using property I(c), reorder the summation with respect to subgraphs of F, and obtain 4.2.2.

Ix

2 0 r 1 2 R ~,~ - ~ v B ( F ) R 0#

r

~,~ =

Z

"yi:lPI subgraphs of F'

U~,(B(~,~) x ~r.s.p. ~ ~ , ~.~,..~.~ ~,~J~, .

(23)

can be expressed as (1 + h O / a h ) l ~ . Moreover, the combinatorial problem being solved as above we get after summation over all graphs F contributing to the Green function (01T(X)IOU°~r:

B(y~)t~

#

z ° (0lY(X)]0)pr°p~r= ~ 2 2 0# 2 n~2 r=0 {il,...,ir} l~ii~_n

l(

~+h

r!

Op~ . . . 0 . @,"

1 ~r.s.p. (0[T(qb(pl)' • • ~ [ ~. e . S"L~10\Pr°per[ )l / lo,-0j1 X (0 W(N[f

dx 1 [ki~__1 (ii]__kO~O)~l](x)]X)

0 ) prOper "

(24) We can use the action principle to express hO/Oh as the DVO ~ dx N [ - i~n(x)]:

~x2 __0 (0[T(X)10)proper = -~ax '°")E E OIx 2

r{ _(--i)r _ ~r.~.,_. 1 ~n

,>=2 r=o {i...... i,} t

l<=ii<=n

r!

--0r Opt'

Opi"/

(0[T({1-~ dx N [ i S ~ e , ( x ) ] } ~ ( p l ) . . • 45(p~))10)°r°perlp,.o/ 1 X (0 T ( N [ I dx 1 [ k~I__1 0~,~)~](x)]S)[0) prOper . (25) 4.3. R e n o r m a l i z a t i o n group equation

Having expressed the differential operator # 2a/btz 2 on Green functions as a linear combination of DVO's, we use the method introduced by Lowenstein [2] to obtain Callan-Symanzik and renormalization group equations. We shall do this explicitly for 1~4 theory but generalization to other theories is straightforward.

G. Bonneau / Zimmermann identities

275

W e introduce the differential vertex operations

• ,~2 A,: i~x~L,~,x)],

. F. ( a ~ ) 2 (x)], ~=i~x~'t'-~

• ~4

(26) Then, 2 0

m Om 2 F~ = - m 2 A 1 F N ,

(27a)

a g ~g FN = --g A3FN,

(27b)

NEw = ( - 2m 2A 1+ 2/12 - 4gA3)Fu,

(27c)

I't"2 ~31.12I'M = (m 24/11 + ~/12+ §gA3)Fu

(27d)

with, f r o m eq. (24)*,

m24 = gTg0 (--El,. r.s.p. (01T(~(0)gS(0))10)proper)

= g

~ ( ' -~t

__1 ~ ;

'2 1 g =m[~ 16rr 2

r.s.p. 2 x 4 0p, b " ( 0 [ T ( q ~ ( p ) ~ ( - p ) ) ]

1{

g

,~2

2\1--~ 2] +'" .]

0>~o~r o)

_ 1 ( g ~2 - ~x-g~/+.-., gg = ( - 1 + g ~ ¢~ g ) ( - ~ t1. r.s.p.

2 0

,.

(0[T(~(0)~5(0),~(0)~5(0))10)oroper)

(28)

(0[T(gS(0)gS(0)cb (o)gS(o))Iow°~*'~

-- g ~ (-~t r.s.p. (0lT(gS(0),~(0)~(0),~(0))10)~o~ I

--~[~ 16~r~ 2 - 3 ( ~g) 2 + ] Because of the four relations (27) a m o n g the three D V O ' s (26) there exists a linear relation b e t w e e n the four left-hand sides of eqs. (27) which can be written as

tz cglzz+ Tm(g)m 2

* For qb4 theory B =

Lin t-

+ fl(g)g

=

V+ 1 = V+ 1-½N and gO/Ogcounts the number of vertices V.

(29)

276

G. Bonneau / Zimmermann identities

where we have used the fact that q, tr and ~ in minimal renormalization depend only on the coupling constant g. We could as well have written:

[ I-t2b + m20 0m2+ /3(g)g o-~--Ny(g)]1"N =(yrn(g)-- l )m2 All'N=--ctm(g)mEA11"N. (30) Here 3 g 1__7( g ~2 / 3 ( g ) = 2 ~ + g = 2 16~r 2 6 \ 1 6 z r 2] + " " 1

v(g)-2

2

-24~16~.2} +" • •, 1

Ym(g)=F+4

g

2 16~-2

5{ g ~ 2 -~\1--~2] + " ' "

(31)

The link with the usual formulas of multiplicative renormalization is straightforward: for instance in qb4 theory if Z~,, Z,, and Zg are the (infinite) renormalization constants (see ref. [12]): ~ R = l(oq ~0)2

1 2--2 (~)04 - 2 m o ~ o - go 4--~-

= ½Z,~(O~(/')2 -

t~ 4

~Z~,Z~m z~z _ gZ~Z~ ~ . .

(32)

Then formulas (28) give 0 (1) [l=g~g(~Z~ +Z~)), I

_

a - g~g (Z~) ±!7(1)~ m 2L-J g P ,

a = - g ~g (~Zg'), where Z ~ ( g ) = ~

Z~)!g).

i=0

(33)

P

Comparison of eqs. (31) and (33) then gives the well-known expressions for/3, y and ym as functions of the renormalization constants: (g) =

a_ z . )

3g g '

=

1

(9 Z ~ ) ,

y,n(g) = g-~g Z~ ) •

(34)

4.4. Generalization to Green functions involving a normal product Nit?8 (x)] The change in the DVO/z20/0/z 2 is similar to the one for Zimmermann identities (see subsect. 3.5) and will generally lead to a mixing between Green functions with

G. Bonneau / Zimmermann identities

+

a)

277

F

b)

Fig. 3. Graphical analysis of the differential vertex operation ix28/Slx 2 on a graph G with an insertion N[68(x)]: (a) normal terms, also present when there is no insertion [eqs. (23), (24)]; (b) extra terms involving new insertions N[6~(x)]. different insertions N [ ~ ( x ) ] of the same q u a n t u m n u m b e r s as ~8 (refs. [12, 11])*. Fig. 3 gives a graphical idea of the origin of the extra terms. T h e o t h e r D V O ' s are not c h a n g e d but the c o u n t i n g identity (27c) is modified in the usual m a n n e r [2]. F o r instance in ~4 theory, if ~78(x) is a p r o d u c t of p fields ~ ( x ) , one has to change N in N - p to get the correct answer. T h e r e n o r m a l i z a t i o n g r o u p e q u a t i o n then follows immediately.

5. Concluding remarks D e s p i t e its successes - - especially in the study of gauge theories - - dimensional renormalization suffers f r o m s o m e lacks. In this w o r k two of t h e m are solved: (i) Z i m m e r m a n n - l i k e identities are shown; * Moreover, in a few cases, there are some special anomalies in the renormalization group equation (see ref. [141).

278

G. Bonneau / Zimmermann identities

(ii) the renormalization group equation is proved in the minimal subtraction scheme, without the use of infinite counterterms in the lagrangian. The results involve residues of the simple poles of Green functions where the overall subtraction has not been done. As we know from Breitenlohner and Maison's work that usual properties (Ward identites, field equations, action principle) also hold for these Green functions (page 29 of ref. [10a]), we should be able to obtain relations between these r.s.p, and then, for instance, show the non-renormalization of the triangle anomaly along the same lines as those of ref. [6]. There are the remaining lacks of this method: no general expression for 3'5 and trace anomalies exists yet. We shall show in a further publication [11] that the techniques developed here also answer these two questions. I am indebted to G. Valent for stimulating discussions.

Appendix A Presence of a factor I.t ~ for each loop integration

Only a few points of ref. [10a] are modified by this change*. First, the proofs of appendix B of BM are unchanged: we briefly recall the results. (a) The singular part of the tH integral in the regularized amplitude for graph H is a polynomial P(qH) of degree tOla in the masses, x/e, and external momenta qla of H. With I on dtH tvB(H)_,OH -- . ~ ) tH

vB(H)

H gv.~ (q.; t_,/~)

) tz vB(H) 1 ( d'°H H - vB(H) ~OH!\dtH gv.~(qH;__t. ~) tH=0 + regular part at v = 0 vB(H)

_/z P(qH) + regular part at v = 0 vB(H)

(h.1)

-

It is straightforward to check that the true singular part is in fact independent of /z[~ (0/0/x)(A.1)[~=o exists]. (b) The singular part of the tH integral in the regularized amplitude for G can be expressed in terms of the same polynomial P(qH) inserted as a vertex function into the regularized amplitude for G / H (lemma 5 of BM): sc~.(n)/x~n,.) (oJ~l d'°"dtnI~G'~ t.=o '~ [¢~IB(.)

~o(H )_

o, G/. Iu - -1B ~ UH(e~{AP(qH))I....

singular part of the tH integral insertion of P(qH) into I °/u (A.2) for G * We do not recall here all the definitionsthat were givenin BM but simplyexpressthe main linesof the proofs.

G. Bonneau / Zimmermann identities

279

Then the convergence proof has to be slightly modified and proposition 3 of BM now reads: consider any X0 ~ ~ (forests of pair-wise disjoint, 1PI genuine subgraphs of G), Xo c c~ and define X = {H' • ~: H' __ H for some H • Xo}. After performance of all subtractions corresponding to subgraphs H • X, the contribution of (~, or) to R ~ , is a sum of terms of the form

HJ-J~\X{ ( ~ - - C H ) f d/xn~'h-'onZ.( - i0/00)}H~Xo{ ~:~l'"gH(/X'm V)} x

(A.3)

gx(Ca, 0; t,/3; OIo 0,

where (_t,~ ) and 0 are scaling variables and scaled u's for G/Xo (i.e., _u, for H E X0 are already set to zero) and Cl are the scaled momenta appropriate for the family c£\X; gH • ] ~ for some K and gx is some element of the abstract algebra of covariants with complex coefficients which are C °o in (_t,_/3), analytic at v = 0 and, due to e > 0, exponentially decreasing as tG -* 00. For completeness - - and understanding of the following proof - - we reproduce here the definitions of functions gH • ] Hx (as well as others needed below) and their properties (lemma 2 of ref. [10a]). Given a maximal forest ~ for G, we define sets of functions J ~ (for K • [0, B(H)[ ) and ] ~ (for K • [0, B(H)]) for all H • ~:

J~={f(~,v):f(~,v)=$ ~

I-I gH,(~,v) w i t h g H , • f ~ % K =

H'ct~(H)

] ~ ={g(~, v): g = f o r ~ - ' f w i t h f • J ~ ;

or g(~:, v ) = I i

~

H'c~(H)

KH,},

f(x, v) with f • J 6 -1 } ;

if B(H) = 1 (i.e.,/z(H) = &) there is only K = 0 and j o = {~:~}. Then lemma 2 of BM says: let f(~:, v) be any element of J ~ ; then (a) (b) (c) (d)

f(~:, v) /] -K ~mH)c.,~:~,. -".m=l with some constants cm; f(~:, 0) = c(Ln ~:)K with some constant c; f(~t, v) = ~ifti (1~,v)f2i(t, v) with fii • J~" such that KIi + K2i = K ; j o = {~=m., m = 1 . . . . B(H)}. =

(A.4)

We now repeat the proof of (A.3) given in BM, with the modifications needed to take into account the introduction of the scale/~. The proof is by induction with respect to ~n~xo B(H) = Ixl. (i) The statement is certainly true for X = ¢.

280

G. Bonneau / Zimmermann identities

(ii) As in BM we start with (~- CH) f d/ZH~-'°~ZH(-- iO/Ofi_)

1-[

H'¢t.t (H)

{~,'°H'gH,(/Z~:H,,V)} gx,l~_.=0

(A.5)

and get (here gH' = gatH = ~n)

(] -- C H )

dt_____n_n fooH tH (~HtH)-'~Hf(~HtHIX,v)g'.~ (tn, fl; (1, _fi) - -

= ~H.I.I (~ _ C H ) - K - 1

. Cm m=l

-

( ~H/d, ) vm

d'~

dtH g ~'* tn=O (A.6)

+fO(~HtH'nf('HtnlZ, P)g'~.~)re~lar}' where f e J nr and we used eq. (A.4a). Here also it is straightforward to check the part (labelled forest by labelled forest):

~

sing

.....

independence oflz of the true singular

,'-" Y c m ( ~ ) m=l

~, - ,

(,OH.

dtH

g~,~

m=0

and f(~:att, p = 0) exists [eq. (A.4b)]. (a) If H ~- G, ~n = 1 and we get, as in BM, the counterterm for H K jB(H) 1

\

m

/

v- - | ~.. -- Cm}T~rP(q), \m=l

(A.7)

and then

cm (A.5)---- B(o)__l ~

_r_l(~ vm-

T~r)P(q) + fo ( )reg,

(A.8)

m = l /~

where T~r is the Taylor operator of degree K in v. This proves theorem l(b) once proposition 3 [eq. (A.3)] is checked. (b) If H ~ G we use eq. (A.2) to express 1

d a'H

oJnl dtn g'l~=o as the amplitude for G / H with polynomial P(qa) inserted as a vertex part, eq. (A.7)

G. Bonneau / Zimmermann identities

281

which gives the counterterm for H, and eq. (A.4c) to study the regular part:

(A.5)- ~.i.i { p-K-1B~2H)c"l[(~H.)vrap_ "=1

m

T~K "P]gG/H

(dttttH'Hf2i(tH,P)g') }

-t-'~ flfl'H/d., /,')f01 i \ tH B(H)

1 [ (~H~./,) v""

m=l

m

reg

-- 1]( T ~P)gG/H

+ v -K-1 s~H)c,. 1 (~:H~)~"([1 _ T~]P)gO/H m=l

m

+ E fl,(¢HZ, V)[/0 ldtHtH "-:---/2j(tH,v)t{ff"(1- T,o,H H )g , J

-fl

dtH -~--HfE~(tH' ~'])tHX---tOH--t°IrI--1 ltH gj~'11

re~"dt ~" r " :~:H°'H{~'--Km~)CmJl,.=I t t (T,P)go/H B (H) 1 + ~.~ C,.~"~"(~HI.t.)v,.v-K-I[(1 - T~K )P]go/H ,.=1 m

+• g~ (~a/X, ~') x gi(reg, at v = 0)} i

-,o f t eH~'dt

7 I(,' v)(T~P)go,,n+Y,k g~(~H~, ~') X gk(reg, at

} O)

= ~H ~°F! ~,, gH(~H/d., //) X g x ( q , 5_; t, [3; ~,)

This proves proposition 3 and then also theorem 1(a). Appendix B Zimmermann identities in momentum space dimensional renormalization (refs. [8, 9])

We shall give here the sketch of the proof, which follows very closely the usual one, using the notations of Collins [8]. There, the renormalized amplitude is given by R r = lim R r,

~-.o R F~ -- E l-I ( - ~'~)If, U ~U

(B.1)

282

G. Bonneau

/ Zimmermann

identities

where U is a forest (not necessarily maximal!) for F and I r is the m o m e n t u m space F e y n m a n amplitude*. ~ I r is I r/v with pole part of I~ inserted as a vertex function. T h e similarity with the Z i m m e r m a n n forest f o r m u l a is obvious but the convergence p r o o f has not b e e n given in this formulation. Yet it is easy to c o m p u t e R r~ where a factor v has been attached to a special vertex V. T h e forest U being divided into three parts: FU: vi ~FU v ¢¢, vi ~ U a n d V ~ v

i,

FU: ai ~F U ¢:~ai ~ U , V ~ a i , andai c F U, FU: b i ~ F U ¢=> b i ~ U and b ~ g { F vU, FU}, we easily get a formula similar to f o r m u l a (9) (using l e m m a 1):

Rr"-uRr=

~

~U

[I

(-~v){r.s.P.v,(

U v' ~ F v v~IFvU,FU}

~

u(-~,)/r)}.

(B.2)

x V~{Fv,FA }

"yq~ Vt

"y~V t

H e r e r.s.p.~, means that one inserts into F l y ~ the r.s.p, of ([I ( - ~ ) I ~ ' ) T h e likeness with the Z i m m e r m a n n f o r m u l a (4.17) of ref. [13] makes the p r o o f straightforward. O n e reorders the s u m m a t i o n over forests U fixed subgraph r of F for which V is a vertex, a forest containing r can

**. the rest of of F: for a be written

U = U~ u {r} u U 2 ,

where U2 ~ {forests for r}, U , ~./g(r) = { y : y ~ T o r

ynr=&}.

As there is a o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n elements of ~ (T) and those of {forests for F/r}, we get r~

r

R~ - ~ n ~ =

(

E

E

H (

_ ~,))

subgraph of r , uE~

{r....rz n

LU2 3'~U2

(B.3) where 0 , m a k e s the insertion into F i r of the polynomial*** r.s.p. R~ as a vertex function. T h e n everything goes as in subsect. 3.3. and we get the same result. * of course internal momenta integrations are to be made to define the pole parts! ** Which is generally not a polynomial in the masses and external momenta of v i but this is not a problem in momentum space formulation! *** As subtractions are done on all 1PI genuine subgraphs of r, we know that the r.s.p, is a polynomial (for example, ref. [9]).

G. Bonneau/ Zimmermannidentities

283

Appendix C

Renormalization group in momentum space dimensional renormalization Here we give the proof of the analogous of formula (23), still by induction with respect to the number of loops of graph F: 2

- ~3 z R~F -½uB(F) R r =

Z

x2-r.s.p.R~)R~

'.

(C.1)

Yi : 1PI subgraphs of F

The induction hypothesis (C. 1) being easily proved for l-loop graphs, we suppose it to be true for N-loop graphs. Let us now consider a 1PI graph F with N + 1 loops. For such a graph (N > 0), there always exists a line C that can be cut leaving a 1PI graph G with N loops, external lines being as usual thrown away to define the graph G. Let {Fr} ({FG}) be the set of forests for F (G). We have {Fr} = {{FG}, {F w FG}} and then

/d,

-3//" - 2 R ~ = Ix

As the pole part - ~ r hypothesis,

[~2o----xO[~ F

(9- ~ r )

j,_r/6 u~ / z .,,r/~ ~ E 1-I ( - g a ~ v ) I Vo V e F G

•

(C.2)

is independent of/x we get, with the help of the induction

2~2

(fj,_r/G vrF/GnG~l~v)= ~vR ~ + ~vB(G)R 1

-F

1

--F

+ ~. f dkr/GlZ"OH(B(H) X ~r.s.p 1 . .#H.I/-r/G]~G/H ........

(C.3)

H_~G

We now take the pole part of this equation: 0=p.p. (:u(S(G)+l)/~r)+

Y~ p.p.{UH(B(H) ×~r.s.p._.~,..~ 1

,

H~_G

= ½vB(F) p.p. R- ,r- ~ B1 ( F ) × r . s . p . R-F ,+

Y~ p . p . { . . . } ,

(C.4)

H~_G

where lemma 1 has been used. Combining eqs. (C.3) and (C.4) we get 2

~ O2 R ,

F

1 F =~,B(F)R~ + H_=GEOH(B(H)x~r.s.p. RH)Rr/H

+B(r) x~.s.p. R-vr,

which is the desired result for N + 1 loop graphs. Then everything goes as in subsects. 4.2.2 and 4.3.

284

G. Bonneau / Zimmermann identities

References [1] W. Zimmermann, 1970 Brandeis Lectures, Lectures on elementary particles and quantum field theory, ed. S. Deser et al., (M.I.T. Press, Cambridge, 1970) vol. 1, p. 397. [2] J.H. Lowenstein, Comm. Math. Phys. 24 (1971) 1. [3] J.H. Lowenstein, Phys. Rev. D4 (1971) 2281. [4] M. Gom~s and J.H. Lowenstein, Nucl. Phys. B45 (1972) 252. [5] G. Bonneau and G. Valent, Nucl. Phys. B140 (1978) 350. [6] J.H. Lowenstein and B. Schroer, Phys. Rev. D6 (1972) 1553; D7 (1973) 1929. [7] G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189. [8] J.C. Collins, Nucl. Phys. B92 (1975) 477. [9] E.R. Speer, 1975 Erice Lectures, ed. G. Velo and A.S. Wightman (D. Reidel, 1976). [10] (a) P. Breitenlohner and D. Maison, Comm. Math. Phys. 52 (1977) 11; (b) P. Breitenlohner and D. Maison, Comm. Math. Phys. 52 (1977) 39, 55. [11] G. Bonneau, Trace and axial anomalies in dimensional renormalization through Zimmermann-like identities, preprint Paris LPTHE 79-26, Nucl. Phys. B, to be published. [12] D.J. Gross, 1975 Les Houches Summer School, Methods in field theory, ed. R. Balian and J. Zinn Justin (North-Holland, Amsterdam, 1976) p. 140. [13] J.H. Lowenstein, 1975 Erice Lectures, ed. G. Velo and A.S. Wightman (D. Reidel, 1976). [14] S. Coleman and R. Jackiw, Ann. of Phys. 67 (1971) 552; K. Symanzik, Comm. Math. Phys. 34 (1973) 7 (appendix D).